A $0-2$ law for cosine families with $\limsup$ to $\infty$

Abstract

For $\left(C(t)\right)_{t\in\mathbb R}$ being a cosine family on a unital normed algebra, we show that the estimate $\limsup_{t\to\infty^{+}}\|C(t) - I\| <2$ implies that $C(t)=I$ for all $t\in\mathbb R$. This generalizes the result that $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. We also state the corresponding result for discrete cosine families and for semigroups.